What sort of exercises that can help me lose belly and arm fat quickly?

well the thing is i can’t avoid eating fruits and white bread and all that because my parents are so stubborn and they make me eat all those things your not meant to eat when your on a diet. (except for junk food like chocolates and stuff). and i don’t have a treadmill or exercise bike at home so i can’t exercise on that and my parents say they don’t have ANY time to take me to the gym or swimming and all.so I’m trying to find out a way to exercise without using any equipment. i want to lose my belly fat, arm fat and my face is really kind of puffy especially the cheeks. how can i lose them all quickly?
do i just walk around the house? because my parents don’t let me walk around on the streets by myself cos they don’t have time to take me. and do i walk at a steady pace or fast?
but doesn’t fruits make you gain weight?